Physica C 194 ( 1992 ) 203-204 North-Holland

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PHYSICA

What is the superconducting order parameter for UPt3? M.R. Norman Materials Science Division, Argonne National Laboratory, Argonne, IL 60439. USA

Received 6 February 1992

The author observed that information inferred from current interpretations of experimental data for the heavy, electron super- conductor UPh constrain the order parameter to be of Ez. symmetry. Possible objections to the current interpretations will be given.

Over the past eight years, a large body of experi- mental data has been collected on the heavy electron superconductor UPh. The data are consistent with an anisotropic order parameter from a non-trivial group representation. A large body of work also ex- ists using phenomenoiogical theory to explain this data. The question is whether there is a sufficient amount of information available to determine the actual order parameter.

We start by looking at various thermodynamic properties. Specific heat data are consistent with an order parameter with line nodes [1]. Moreover, transverse ultrasound data indicate that the nodal line is orien!ed perpendicular to the c-axis [2]. Re- cently, this has been confirmed by both penetration depth measurements [ 3] and by thermal conductiv- ity [4]. Moreover, the latter two experiments indi- cate that in addition to the nodal line along the equa- tor, nodal points occur along the poles (the axis being along c). There are two order parameters consistent with such a nodal structure, Elg and E2,,. E~g trans- forms as k:(k~+ik,,) and E2u as kz(kx_+ik.v)2z where z indicates the direction of the d vectnr (d.S=O, where S is the spin vector of the Cooper pairs). The most general from for E2u , o f co_urse~ contains com- ponents with x and y. If these are present, then such an order parameter would not be consistent with the above data.

Next, UPt3 has a complicated phase diagram in the H, T plane indicating the presence of three dif- ferent superconducting phases [5,6]. Such a phase diagram is most easily explained if the order param-

eter comes from a two-dimensional group represen- tation [ 7 ]. Thus, the different phases correspond to different orientations of a vector in a 2D space. We note that the above mentioned order parameters, E ig and E2u, are two-dimensional group representations.

Finally, there is an unusual anisotropy present in the upper critical field data [ 8 ]. At low temperatures (T/Tc~0.4) , the He2 phase lines for Hlic and H ex- planation offered for this behavior is 1hat Pau;i lim- iting occurs for tt[[c and not for H ' c . This would only be consistent with an odd partly order param- eter with a d vector locked to the c-axis [ 9 ]. The E2u order parameter mentioned above has this property. Thus. based on the above interpretations, only E~,, is consistent with the data.

We turn now to possible objections to these inter- pretations. First, the nodal structure of the gap is es- timated from the power law behavior of the tem- perature variation of various quantities such as the specific heat. In reality, the power law behaviors should only exist for very low ratios of T~ Tc whereas experiments have been done in a temperature range above about 0.2T/Tc. Thus, there is always a ques- tion whether the nodal structure is actually lines or points~ although the consistency of all the thermo- dynamic data among themselves would argue favor- ably towards the line nodes plus point nodes inter- pretation. Another point is that a variety of specific heat measurements tend to extrapolate ,,-' ,.,,.. . . . . tem- perature to a finite value of C~ 7.. implying either a change in power law at very low temperatures or gapless behavior [1,5]. The extrapolated value is

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204 M.R. Norman / Order parameter for UPt ~

simple dependent, thus complicating its interpreta- tion. A related effect is the non-observation of de Haas-van Alphen oscillations in the normal state for field directions near the c-axis [ l 0 ]. This implies that either the quasiparticle mass renormalization or the scattering rate strongly increases as the field direc- tion rotates towards the c-axis. For fields along c, the extremal orbits would actually correspond to the equatorial nodal lines discussed above. The leads to the question of whether the nodal line is due to the group symmetry of the order parameter, or due to gap suppression caused by the normal self-energy. Further de Haas-van Alphen measurements will be necessary to look into this point.

Although the phase diagram of UPt3 in the H, T plane is most easily described as due to the degen- eracy of an order parameter from a two-dimensional group representation being lifted by both a weak, in- trinsic magnetic moment and by an external mag- netic field, other models have been proposed: a near degeneracy between two different group represen- tations [ 11 ] and the lifting of the spin degeneracy of a one-dimensional odd parity representation [ 12]. The argument against the former is that the splitting of the phase lines disappears at the same pressure as the intrinsic weak magnetic moment disappears (which in the two-dimensional group representation model is responsible for the splitting) [ 13 ]. The ar- gument against the latter is that the magnetic sus- ceptibility is already highly anisotropic in the nor- mal state, which in turn implies that the assumed degeneracy is not there to begin with.

The crossing of the He2 phase lines (for HIIc and H_kc) at low temperatures seen from resistive mea- surements [8] has also been observed in some ul- trasound experiments (see Adenwalla et al. [6 ] ). Other ultrasound experiments, though, do not ob- serve such a crossing (Thalmeier et al. [6] ). Since the interpretation of the crossing is most easily ex- plained by an odd parity state, it is of some impor- tance that this matter be investigated further.

Finally. microscopic calculations of the order pa- rameter of UPt3 based on spin fluctuation pairing are not supportive of either an E2u or an EI~ state [ 14]. It is difficult to imagine how in the context of these theories such a state could be stabilized.

In conclusion, current interpretations of experi- mental data on UPt~ constrain the order parameter

to be E2u symmetry. Several valid objections to these interpretations have been discussed, though, indi- cating the need for further experimental and theo- retical work before the true nature of the order pa- rameter is unambiguously determined.

Acknowledgement

This work was supported by the US Department of Energy, Office of Basic Energy Sciences, under Contract No. W-31-109-ENG-38.

References

[91 [10]

Ill]

[~21 [131

[14]

[ 1 ] A. Sulpice et al. J. Low Temp. Phys. 62 ( i 986 ) 39. [2] B.S. Shivaram, Y.H. Joeng, T.F. Rosenbaum and D.G.

Hinks, Phys. Rev. Left. 56 (1986) 1078. [3] C. Broholm et al., Phys. Rev. Lett. 65 (1990) 2062;

W.o Putikka, P.J. Hirschfeld and P. Wolfle, Phys. Rev. B 41 (1990) 7285.

[4] K. Behnia el al., J. Low Temp. Phys. 84 ( 1991 ) 261. [ 5 ] R.A. Fisher et ai., Phys. Rev. Lett. 62 ( 1989 ) 1411;

K. Hasselbach, L. Taillefer and J. Flouquet, Phys. Rev. Lett. 63 (1989) 93.

[6] V. Muller et al. Phys. Rev. Lett. 58 (1987) 1224; A. Schenstrom et al., Phys. Rev. Lett. 62 (1989) 332: G. Bruls et ai., Phys. Rev. Lett. 65 (1990) 2294: S. Adenwaila et al., Phys. Rev. Lett. 65 (1990) 2298: P. Thalmeier et al., Physica C 175 ( 1991 ) 61.

[ 7 ] G.E. Volovik. J. Phys. C 21 ( 1988 ) L221: R. Joynt, Supercond. Sci. Tcch. 1 (1988) 210; D.W. Hess, T.A. Tokuyasu and J.A. Sauls, J. Cond. Matter Phys. 1 (1989) 8135; K. Machida, M. Ozaki and T. Ohmi, J. Phys. Soc. Jn. 58 (1989) 4116; R. Joynt, Europhys. Lett. 16 ( 1991 ) 289.

[ 8 ] B.S. Shivaram, T.F. Rosenbaum and D,G. Hinks, Phys. Rev. Lett. 57 ( 1986 ) 1259; B.S. Shivaram, J.J. Gannon Jr. and D.G. Hinks, Phys. Rev. Lett. 63 (1989) 1723. C.H. Choi and J.A. Sauls, Phys. Rev. Lett. 66 ( 1991 ) 484. L. Taillefer and G.G. Lonzarich, Phys. Rev. Lett. 60 ( 1988 ) 1570. R. Joynt, V.P. Mineev. G.E. Voiovik and M,E, Zhitomirsky, Phys. Rev. B 42 ( 1990 ) 2014. K. Machida and M. Ozaki, Phys. Rev. Leu. 66 ( 1991 ) 3293. L. Taillefer et al., J. Magn. Magn. Mater. 90-91 ( 1990 ) 623: T. Trappmann, H. v. Lohneysen and L. Taillefer, Phys. Rev. B43(1991) 13714. M.R. Norman, Ph~rs. Rev. B41 (1990) 170: ibid., 43 ( 1991 ) 6121.